The Cutkosky rule of three dimensional noncommutative field theory

in Lie algebraic noncommutative spacetime

Yuya Sasai (Yukawa Institute for Theoretical Physics, Kyoto University)

in collaboration with N. Sasakura (YITP)JHEP 0906, 013 (2009) [arXiv:0902.3050]

基研研究会「場の理論と弦理論」 July 9, 2009

1. Introduction

Why noncommutative field theory?

• Effective field theory of open string theory

in constant B background

: an antisymmetric constant )(

Seiberg, Witten (1999), etc

Quantum spacetime Planck scale physics

• Classical solutions of matrix modelsIshibashi, Kawai, Kitazawa, Tsuchiya (1996),etc

However, it is not clear that these noncommutative space arise from quantum gravity effect.

• Effective field theory of Ponzano-Regge model with

which spinless massive particles are coupledFreidel, Livine (2005)

In three dimensions, a massive particle is represented as a conical singularity.

The 3D noncommutative field theory might give a field theory of conical singularities.

Deser, Jackiw, ‘t Hooft (1984)

• Not depend on globally flat background space!

• Related with 3D gravity

Ponzano-Regge model

• Partition function of 3D pure gravity with

• 3D lattice gravity theory

• Triangulate 3D manifold with tetrahedra

: SU(2) spin

: Dimension of spin-j rep.

Ponzano, Regge (1968)

particle insertion

: Character of

Freidel, Louapre(2004)

Inserting spinless massive particles on the graph ,

After some manipulations,

propagator vertex

: SO(3) group element

Star product

Freidel, Livine (2005)

Except the propagators, this expression has the same structure as Feynman amplitudes of noncommutative scalar field theory, whoseaction is

where the scalar field is defined by

.

Commutation relation of coordinates is found to be

.

The three dimensional noncommutative field theory is derived fromthree dimensional quantum gravity theory with particles.

We can construct the Lorentzian version of this noncommutative field theory.

Naively, we can expect that the 2+1 dimensional noncommutative field theory will represent the 2+1 dimensional quantum gravity theory with which massive particles are coupled?

Imai, Sasakura (2000)

How about the Lorentzian case?

Noncommutative field theory in the Moyal plane

breaks the unitarity due to the existence of the nonplanar amplitudes if the timelike noncommutativity does not vanish.

However, if we impose the braiding, which is a kind of nontrivial statistics, the nonplanar amplitudes becomes the same as the corresponding planar amplitudes if they exist.

with braiding

( : antisymmetric constant)

Gomis, Mehen (2000)

Oeckl (2000), Balachandran, et.al (2005), etc

However,

Unlike the Moyal case, even the planar amplitudes are nontrivial because of the SL(2,R)/Z_2 group momentum space.

It is a nontrivial issue to check the unitarity in the noncommutative field theory in the Lie algebraic noncommutative spacetime.

• We have checked the Cutkosky rule of the one-loop self-energyamplitudes in the noncommutative theory with the braiding.

• As a result, this theory will not be unitary even when we impose the braiding.

• This is caused by the periodic property of the SL(2,R)/Z_2 group momentum space.

In our paper,

Contents

1.Introduction

2. 2+1 dimensional noncommutative field theory in

3. One-loop self-energy amplitude of the noncommutative

theory and the Cutkosky rule

4. Summary

2. 2+1 dimensional noncommutative field theory in

This noncommutative field theory possesses group momentum space if we assume the commutative momentum operators and the Lorentz invariance.

Imai, Sasakura (2000)

• Lorentz covariance• Jacobi identity

Start with

up to , where is an arbitrary function.

,

(i,j,k=0,1,2)

are identified with ISO(2,2) Lie algebra,

with the constraint

SL(2,R)/Z_2 momentum space

ISO(2,2) Lie algebra

where

• The

where

with

Action of the noncommutative theory in the momentum representation is

This noncommutative field theory possesses a nontrivial momentumconservation (Hopf algebraic translational symmetry) due to the group momentum space at classical level.

To consider the identification , we impose

where . Imai, Sasakura(2000), Freidel, Livine (2005)

In order to possess the nontrivial momentum conservation in the noncommutative field theory at quantum level, we have toimpose the braiding on fields such that

: braiding map

Freidel, Livine (2005), Y.S., Sasakura (2007)

3. One-loop self-energy amplitude of the noncommutative

theory and the Cutkosky rule

Propagator:

3-point vertex:

Braiding:

Feynman rules

One-loop self-energy diagrams in the noncommutative theory

Planar

Nonplanar

Same as the planar!

where

• Since SL(2,R) group space is equivalent to AdS_3 space, we can take

For simplicity, we set .

In the center-of-mass frame, we can take the external momentum as

As a result of calculation,

where

Considering ,

it is enough to take the rangeof as

for the positive energy external leg.

Imai, Sasakura (2000)

Cutkosky rule of the one-loop self-energy diagram

Cutkosky rule

if

otherwise

=

0

Physical process

Unphysical process!

In the range of , the RHS of theCutkosky rule will vanish.

Physical interpretation

In fact, we have shown by explicit calculations that

0

if

otherwise

Thus, the Cutkosky rule is satisfied in the shadow region.

is only allowed in any !

However, this result does not imply that the theory is unitarywhen the mass is smaller than .

Let us consider multi-loop amplitudes.

The Cutkosky rule will be violated when is in the positive energy regions.

If we consider more complicated diagrams, we will see thatthe Cutkosky rule is violated for any values of the mass.

Even if , the Cutkosky rule will beviolated.

4. Summary

• We have checked the Cutkosky rule of the one-loop self-energyamplitudes in the noncommutative theory with the braiding.

• We have found that the Cutkosky rule is satisfied at the one-loop level when the mass is less than .

• However, this theory will not be unitary because if we considermore complicated diagrams, the Cutkosky rule will be violated for any values of the mass.

This disastrous result is caused by the periodic property of the SL(2,R)/Z_2 group momentum space!

What should we learn from this result?

The extension of the momentum space to the “universal coveringof SL(2,R)” may remedy the unitarity property of the theory.

If the 2+1 dimensional noncommutative field theory gives a 2+1 dimensional quantum gravity theory with massive particles,why isn’t the arbitrary negative energy included? Y.S., Sasakura (2009)

Deficit angle =

One-particle solution in 2+1 dimensional Einstein gravity

can be arbitrary negative!

“Universal covering of SL(2,R) group momentum space” will possessarbitrary negative energy.

This extension may give a hint of 2+1 dimensional quantum gravity with massive particles.

Deser, Jackiw, ‘t Hooft (1984)

By the way,